How to find the least upper bound on the van der Waerden Number $W(r,k)$ that is some integer Power of the coloring Integer $r$

Abstract

What is a least integer upper bound on van der Waerden number $W(r,k)$ among the powers of the integer $r$? We show how this can be found by expanding the integer $W(r,k)$ into powers of $r$. Doing this enables us to find both a least upper bound and a greatest lower bound on $W(r,k)$ that are some powers of $r$ and where the greatest lower bound is equal to or smaller than $W(r,k)$. A finite series expansion of each $W(r,k)$ into integer powers of $r$ then helps us to find also a greatest real lower bound on any $k$ for which a conjecture posed by R. Graham is true, following immediately as a particular case of the overall result.

25 pages, one Table, no figures. Small addition to Section 1. Slight revision of Theorem 3.1, Section 2. Typo correction for Theorem 7.2, Section 7, where \(\frac{k}{r} = o(1)\) ought to have been \(\frac{k}{r^{n}} = o(1)\) and \(k \ll r^{n}\)

References in corpus (1)

• Some New Exact van der Waerden Numbers